Page 156 - Schaum's Outlines - Probability, Random Variables And Random Processes
P. 156
CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 149
4.41. Verify Eq. (4.36).
Since X and Y are independent, we have
The proof for the discrete case is similar
4.42. Let X and Y be defined by
X = cos 0 Y = sin O
where O is a random variable uniformly distributed over (0, 271).
(a) Show that X and Y are uncorrelated.
(b) Show that X and Y are not independent.
(a) We have
P
(0 otherwise
Then E(X) = I:m~fx(x) di = l* cos Ofe(0) d0 =
Similarly, E(Y)=~ $ sin 0 d0=0
ax px
1
E(XY) = i; cos 0 sin 0 d0 = - sin 20 d0 = 0 = E(X)E(Y)
Thus, by Eq. (3.52), X and Y are uncorrelated.
1 1 1
2
sin2
E(Y2) = Z; In 0 d0 = ['(I - cos 20) d0 = -
1
(
8
E(x'Y') = & % cos2 0 sin2 0 10 = - 1 - cos 4, d0 = -
Hence
If X and Y were independent, then by Eq. (4.36), we would have E(X2Y2) = E(X2)E(Y2). Therefore, X
and Y are not independent.
4.43. Let XI, . . . , Xn be n r.v.3. Show that
n n
Var xaiXi = x xaiajCov(Xi, Xj)
(i11 ) i=1 j=1
